Constant rate PCPs for circuit-SAT with sublinear query complexity

The PCP theorem says that every NP-proof can be encoded to another proof, namely, a probabilistically checkable proof (PCP), which can be tested by a verifier that queries only a small part of the PCP. A natural question is how large is the blow-up incurred by this encoding, i.e., how long is the PCP compared to the original NP-proof. The state-of-the-art works of Ben-Sasson, Sudan and Dinur show that one can encode proofs of length n by PCPs of length (n poly log n) that can be verified using a constant number of queries.

Here we show that if the query complexity is relaxed to n^0.1, then one can construct PCPs of bit-length O(n) for circuit-SAT, and PCPs of bit-length O(n log n) for any language in NTIME(n). More specifically, for any epsilon>0 we present (non-uniform) probabilistically checkable proofs (PCPs) of bit-length O_epsilon(n) that query n^epsilon proof-bits for circuit-SAT instances of size n. Our PCPs have perfect completeness and constant soundness. This is the first constant-rate PCP construction that achieves constant soundness with nontrivial query complexity (o(n)).

Our proof replaces the low-degree polynomials in algebraic PCP constructions with tensors of transitive algebraic geometry (AG) codes. We show that the automorphisms of an AG code can be used to simulate the role of affine transformations which are crucial in earlier high-rate algebraic PCP constructions. Using this observation we conclude that any asymptotically good family of transitive AG codes over a constant-sized alphabet leads to a family of constant-rate PCPs with polynomially small query complexity. Such codes are constructed in the appendix to our paper for the first time for every message length, after they have been constructed for infinitely many message lengths by Stichtenoth.

Joint work with Swastik Kopparty and Or Meir, with an Appendix by Henning Stichtenoth.

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